Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Fourier Transform of $f(t+a)$ if $f(t)$ has tranform $F(k)$?

I know the formula $$f(t) = \int^{+\infty}_{-\infty} F(k)e^{ikt} \, dk$$ and I've seen that for computing $f'(t)$ it's a case of differentiating $e^{ikt}$ inside the integral, so $f'(t)=ikF(k)$ Can ...
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56 views

Fourier transform of n-th power of autocorrelation of a random process

I'm having troubles in understanding how Fourier transform of the n-th power of a time function is obtained. In particular I came across to a particular result with respect to the calculation of the ...
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0answers
553 views

Using the Modulation property of the Fourier Transform

I'm working on a problem: Let $X(w)$ be the Fourier transform of $x(t)$. Find the transform of $y(t)=x(5t+3)\sin(2t)$ in terms of X(w). I am table to take the Fourier transform of $x(5t+3)$ and ...
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1answer
47 views

Fourier Transform of a conjugated function (in the fourier plane)?

I'm trying to figure this out. Assume lower case is in $(x,y)$ and upper case is in $(fx,fy)$. I understand 2 forward FFT's lead to a sign reversal. FFT$(t(x,y)) = T(fx,fy)$, FFT$(T(fx,fy)) = ...
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26 views

How to see this step in deriving an equality in Fourier series?

(Previous steps are omitted.) By convergence of Fourier series, we have $$ \frac{\pi}{4}+\sum_{n=1}^{\infty}\frac{(-1)^n-1}{\pi n^2}(-1)^n=\frac{\pi}{2} $$ Then how come we can get this from the ...
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1answer
34 views

Real-valued Discrete Fourier Transform

I have sequence of $N$ real numbers: $\mathbf{x} = (x_0, x_1, \ldots, x_{N-1})$. Discrete Fourier Transform (DFT) is defined as $$ X_k = \sum_{n=0}^{N-1} x_n e^{-i 2\pi k \frac{n}{N}}, \quad ...
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18 views

Let $f(x) := |\cos \frac x 2 |$ for $x \in \mathbb R$. Show by using Eulers formula that the $n$'th Fourier coefficient $c_n$.

Let $f(x) := |\cos \frac x 2 |$ for $x \in \mathbb R$. Show by using Eulers formula that the $n$'th Fourier coefficient $$c_n = \frac 1 {2\pi} \frac {(-1)^{n-1}} {n^2-\frac 1 4}$$ with respect to the ...
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23 views

Let $f(x):= \sum^{\infty}_{-\infty} d_n e^{inx}$ for $x \in \mathbb R$. Find the Fourier series for $f$ and show uniform convergence against $f$.

Let $\sum^{\infty}_{-\infty} |d_n| < \infty$ and define $f(x):= \sum^{\infty}_{-\infty} d_n e^{inx}$ for $x \in \mathbb R$. Find the Fourier series for $f$ and show it converge uniformly on ...
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1answer
42 views

Fourier series of $f(x)$ and its convergence

sorry for the inappropriate format. I'll edit asap. question is about convergence of series. Can you explain why is "f(x)=1 ,f(x)=1/2" please.
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148 views

Inverse Fourier transform using Residues for a ratio of hyperbolic functions.

I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have $$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I ...
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1answer
39 views

Enlarging the space $PC(a,b)$ to include functions with one or more infinite singularities

I'm reading a Fourier analysis book and on the chapter about convergence and completeness of orthogonal sets of functions I have one part which I don't understand. I have uploaded the part as an image ...
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1answer
99 views

Fourier Transform of x(-2t+4)

How would one go about obtaining the Fourier Transform of a general signal $x(-2t+4)$? I know one can use a table of Fourier Transform properties to easily evaluate this, but I want to use the ...
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1answer
56 views

Fourier transform and sufficient condition…

Does anyone could give me a sufficient condition on $f$ so that the Fourier transform of $f$ (denoted as $\hat{f}$) is in $L^{1}(\mathbb R)$. The Fourier transform here is the linear operator ...
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58 views

Confused about isolating an argument from a fraction and computing a value for it

Ok so if I have a fraction which looks as follows $$\frac{8π}{N + 1}$$ How could I calculate N on its own? The above fraction is the mainlobe width formula for a Hamming window and I'm trying to ...
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1answer
470 views

Computing 2d radially symmetric Fourier transforms (with Wolfram Alpha)

Let's assume that I "know" $\mathcal{F}\{\operatorname{circ}(r)\}(\rho)=\frac{J_1(2\pi\rho)}{\rho}$ $\mathcal{F}\{(1-r^2)\operatorname{circ}(r)\}(\rho)=\frac{J_2(2\pi\rho)}{\pi\rho^2}$ because I ...
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494 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
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1answer
66 views

Need help on clarifying a step in proof of Plancherel's theorem in Evans' book PDE

I was stuck when I read the proof of Plancherel's theorem in Line 9, Page 188 of Evans' book Partial Differential Equations, 2nd Edition. Evans wrote there (I quote here): $~~~~\text{2. }$ Now take ...
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113 views

Using Fourier Transforms to Solve the Heat Equation PDE In Infinite Three Dimensions

Problem: Using Fourier transforms, solve for $u(x,y,z,t)$, where $$u_t=D\nabla^2 u$$ $$ -\infty<x,y,z<\infty,t>0$$ $$D>0, u(x,y,z,0)=f(x)f(y)f(z)$$ and $u\rightarrow 0$ as ...
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17 views

How to solving for t in the first order derivative of a Fourier Synthesis.

How can I solve for $t$ given, $$ 0 = \frac{2\Pi}{T} \sum_{k=1}^N k(a_k cos(2\Pi\frac{k}{T}t) - b_k sin(2\Pi\frac{k}{T}t)) $$ What I am trying to do is, given a complex wave in the form of a Fourier ...
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1answer
84 views

Computing Fourier transform of a surface measure.

I am almost beginner in the topics of Fourier transform. So, I am asking this question here. Let $n=3$ and let $\mu_t$ denote surface measure on the sphere $|x|=t$. Then how do we show that $$ ...
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51 views

2D Laplace Equation with Dirichlet Condition

I'm having trouble following the solution for this question. Firstly (underlined in orange) why is the general solution of this form. I would have thought it would be the more general ...
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1answer
467 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
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1answer
166 views

Fourier Transform Identity (Convolution & Symmetry rule)

I don't follow the part of the solution highlighted in green. Use the symmetry rule to show that $${\cal F}\{f(x)g(x)\}=\dfrac1{2\pi}\left(\hat f(\omega)*\hat g(\omega)\right).$$ Convolution ...
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137 views

Heisenberg uncertainty principle in D-dimensional

For Heisenberg uncertainty principle in D-dimensional there is $d^2$ in the formula.where does this additional term comes compared with the case of one dimensional?
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1answer
208 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
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2answers
112 views

Dirac's delta integration

What about the following integral? $$\int_0^a x^3 \delta(x-1) dx$$ If $a$ is more or less than 1 it's all clear, but what if $a=1$. Is the integral is equal to $1/2$ ? Edit: this is my motivation, ...
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123 views

Good family of kernels in $\mathbb{R}^n$

I'm trying to prove that, given the heat equation $u_t = \Delta u$ with boundary values $u(x,0) = f(x)$, the solution given by $$u(x,t) = f \star H_t^{(d)}(x)$$ is continuous up to the boundary ...
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1answer
73 views

Most computationally efficient way to find convolution of a matrix kernel with impulse response?

Let say if we wish to filter an input sequence x[n1, n2, n3] of NxNxN points using an Linear Shift Invariance system with impulse response h[n1, n2, n3], where the filter is a separable sequence, ...
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1answer
38 views

pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?

Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
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328 views

Fourier Transform on compact groups

I'm trying to get my head around the concept of Fourier Transform on a compact group. The standard definition is $$\widehat{f}(\pi)=\int_Gdg\,f(g)\pi(g)$$ where $\pi\in\widehat G$, the Pontryagin ...
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1answer
358 views

abs(x)cos(x) in Fourier space

I am working on some problems concerning Fourier Transform and I am facing something I don't understand. I am trying to understand what is the representation of the function f(x)=abs(x)cos(x) in the ...
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26 views

$f\mapsto \sum_{n\in \mathbb Z} |\widehat{F(f)}(n)|$ lower semi continuous?

Let $T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
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333 views

Trigonometric identity involving sum of “Dirichlet kernel like” fractions

Computing the eigenvalues of a matrix related to fast Fourier transform, we stumbled upon the following identities. Let $k$ and $N$ be natural numbers with $k<N$, then: $$\sum\limits_{j=1}^N ...
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1answer
298 views

Integrate $\int_{-\infty}^{\infty}\exp\left(-\frac{\pi^2t(2x+1)^2}{2c^2}\right)\cos\left(\frac{(2x+1)\pi y}{c}\right)\exp(-2\pi i kx)dx$

By the poisson summation formula we have: $$\frac{1}{c}\sum\limits_{k=-\infty}^{\infty} \exp\left(-\frac{\pi^2t(2k+1)^2}{2c^2}\right)\cos\left(\frac{(2k+1)\pi ...
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1answer
103 views

How to calculate the Fourier Transform of a constant?

The definition of the FT in engineering is: $$\int_{-\infty}^{\infty}f(x)e^{-j2\pi ft}dt$$ I'm having trouble calculating the FT of a constant, such as $\frac{1}{2}$: ...
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36 views

How do I tackle this integral: $\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$? Is my solution correct?

I want to solve the following integral: $$\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$$ I did the following: Substitute $\gamma(k) = k-k_0 \Leftrightarrow k = \gamma + k_0;~\gamma(\pm\infty) = ...
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2answers
106 views

Fourier Transform: why do all segments generate the same magnitude response?!

I'm working on a DTMF program, and what I've done is to break the one long input signal I initially receive into a bunch of smaller components. I perform an FFT on each of the small components and ...
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1answer
221 views

Multiply a circulant matrix by a vector with FFT.

I am asked to write a Matlab program to find the coefficients of the resulting polynomial which is the product of two other polynomials. However, I need someone to clarify the underlying concepts for ...
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1answer
67 views

Holomorphic Schwartz-space-valued function

I was trying the last few days to generalize the Paley-Wiener theorem in a quite obvious direction... or so I thought. The original Paley-Wiener theorem talks about functions and distributions with ...
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1answer
187 views

An issue of applying Fubini's theorem in Fourier transform on Schwartz space.

Let $\hat{f}$, $\hat{g}$ be the the Fourier transform of $f$ and $g$ respectively where $f$ and $g$ are the members of the Schwartz space $\scr{S}{(\mathbb{R}^{N})}$. Then in the process of ...
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1answer
32 views

Apply the Fourier Transform to $A\cdot e^{-a|k - k_0|}$

I have the following problem: The task is to show that $$f^*(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(k) e^{ik(x-vt)} dk$$ with $f(k) = A\cdot e^{-a|k - k_0|}$ equals $$f^*(k) = ...
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1answer
146 views

Proving that the function set $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set

I have the the following problem from my Fourier analysis book: Show that $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set in $PC(0,l)$, i.e. class of piecewise ...
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1answer
92 views

Not understanding one step in derivation of Dirichlet kernel

I was reading some notes on the Dirichlet Kernel and they have a proof of how it reduces to $\sin(2\pi(N+ 1/2)t)/\sin(\pi t)$. I could follow the steps except for one early step which is the ...
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47 views

d-dimensional integral of exponential and determinant

I'm working on a question from Stein and Shakarchi's Fourier Analysis. This is exercise 5 from chapter 6: Let $A$ be a $d \times d$ positive definite symmetric matrix with real coefficients. Show ...
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80 views

Method of PDE solution by Fourier transform

In Rudin's Functional Analysis (Chapter 7, exercise 17), Rudin claims that for $n=1$ or $2$, if $u$ is a distribution on $R^n$ with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded ...
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147 views

What is the Fourier series of $e^{\mu\cos\theta}$?

Motivation: I want to solve this convolution problem on the circle: find $f$, given $g$ and $$ g(\theta) = \int_{S^1} e^{\mu\cos(\theta-\phi)}g(\phi)\ d\phi. $$ To do this, I want to find the Fourier ...
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1answer
24 views

why is integral of dirchlet from negative pie to pie equal to pie

Hi : I'm reading a text called "Fourier Transformations For Pedestrians " and it's a nice book. But I am stuck on the following. On page 34, the author states that $\int_{-\pi}^{\pi} D_{N}(x) dx = ...
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2answers
259 views

What's the point of Dirac delta function?

I have heard that The main useful property of Dirac delta function is it's fundamental property that $$ \int_{-\infty}^{\infty}f(x)\delta(x-a)dx=f(a) $$ I don't understanding why this equation is ...
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1answer
171 views

Why divide by N (length of input sequence) during IDFT?

During DFT of a input sequence of length N, we find X(k). We find inner product with a basis vector to get the coefficient: X(k) = <x[n], e[k, n]>    |  k = 0, 1, 2, ... ...
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1answer
35 views

Spectrum of a Rectangular signal?

I would be grateful if you can help me with i) ii) and iii) especially with ii and iii